As far as the asthma goes, I also have asthma, and I have never breathed so hard with my heart pounding so fast as on my bike going up a steep long hill. (For comparison I jog, do aerobics, snowshoe, and hike at high altitudes). Luckily my asthma is easily controlled with my inhaler so I don't consider it a set back. I would say that cycling can be as easy or hard on the lungs as you want it to be, depending on where you ride, how long you ride, and how hard you ride.
I have women's specific road bike, Trek 1500 WSD, which fits me well. Its all about the fit and the only way you'll know which bikes work for you is to go test ride them. For some women, women's specific just doesn't work. Good luck.
The best part about going up hills is riding back down!